Central Limit Theorem
Why
The (normalized) sum of several independent and identically distributed random variables tends toward a normal distribution.1
Result
Let $(X, \mathcal{A} , \mu )$
be a probability space.
Let $\seq{f}$ be a sequence
of independent and identically
distributed real-valued
random variables on $X$
with
$\E (\seqt{f}) = \mu < \infty$
and
$\var(\seqt{f}) = \sigma ^2 < \infty$
for all $n$.
Define $s_n = \sum_{i = 1}^{n}f_i$.
For all real numbers $t$,
\[ \lim_{n \to \infty} \mu \Set*{ x \in X }{ \frac{ s_n(x) - n\mu }{ \sigma \sqrt{n} } \leq t } = \Phi (t). \]
- Future editions may modify this statement and further specify the word “tends.” ↩︎